This edition has been called `startlingly up-to-date', and in this corrected second printing you can be sure that it's even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.
Problems and Tricks.- Number Theory.- Some Applications of Elementary Number Theory.- Ideas and Theories.- Induction and Recursion.- Arithmetic of algebraic numbers.- Arithmetic of algebraic varieties.- Zeta Functions and Modular Forms.- Fermat's Last Theorem and Families of Modular Forms.- Analogies and Visions.- Introductory survey to part III: motivations and description.- Arakelov Geometry and Noncommutative Geometry (d'apres C. Consani and M. Marcolli, [CM]).
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Softcover reprint of hardcover 2nd ed. 2005